How do you simplify #[2x(x+6)^4-x^2(4)(x+6)^3]/(x+6)^8#?

Answer 1

Hazel Anglin

Expression #= (-2x(x-6))/(x+6)^5#
#x!=-6#

Expression #=(2x(x+6)^4-x^2(4)(x+6)^3)/(x+6)^8#

#=(2x(x+6)^4-4x^2(x+6)^3)/(x+6)^8#

#=(2x(x+6)^3*(x+6-2x))/(x+6)^8#

#=(2x(x+6)^3*(-x+6))/(x+6)^8#

#=(-2x(x+6)^3*(x-6))/(x+6)^8#

#=(-2xcancel((x+6)^3)*(x-6))/(x+6)^(8-3)# [#x!=-6#]

#=(-2x(x-6))/(x+6)^5#

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Answer 2

Isaiah Anthony

To simplify the expression [2x(x+6)^4-x^2(4)(x+6)^3]/(x+6)^8, you can follow these steps:

Expand the terms inside the brackets: [2x(x^4 + 4x^3(6) + 6^2x^2(4) + 6^3x(4) + 6^4) - x^2(4)(x^3 + 3x^2(6) + 3^2x(6^2) + 6^3)] / (x+6)^8
Simplify the expanded expression: [2x^5 + 48x^4 + 288x^3 + 1152x^2 + 20736x - 4x^6 - 72x^5 - 432x^4 - 1728x^3] / (x+6)^8
Combine like terms: (-2x^6 - 24x^5 - 144x^4 + 288x^3 + 1152x^2 + 20736x) / (x+6)^8

Therefore, the simplified expression is (-2x^6 - 24x^5 - 144x^4 + 288x^3 + 1152x^2 + 20736x) / (x+6)^8.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.